How does adding a small change to an neuron's weighted input affect the overall cost?
I believe he's just saying that:

$$
\frac{\partial C}{\partial z_j^l} \Delta z_j^l \approx \frac{\partial C}{\partial z_j^l} \partial z_j^l \approx \partial C
$$

so that the change in cost function can be arrived at simply for a small enough perturbation $\Delta z_j^l$.

Or, taking that line of approximations backwards, the change in the cost function for a given perturbation is just:
$$
\partial C \approx \frac{\partial C}{\partial z_j^l} \partial z_j^l \approx  \frac{\partial C}{\partial z_j^l} \Delta z_j^l
$$